Advances in Differential Equations

The subcritical motion of a surface-piercing cylinder: existence and regularity of waveless solutions of the linearized problem

Dario Pierotti

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We consider a boundary value problem in a strip, which is obtained by linearization of the two-dimensional problem of a stream about a slender cylinder, semisubmerged in a heavy fluid of finite depth; it is assumed that the cylinder has uniform, subcritical speed in the direction orthogonal to its generators. We discuss in particular the waveless statement of the problem, characterized by the asymptotic condition of vanishing flow oscillations at infinity. By suitable variational formulations of the problem, we find two classes of solutions, differing by their regularity properties. The most regular solutions exist for particular shapes of the cylinder's section and provide a velocity field which is everywhere continuous and bounded in the strip. The solutions of the other class exist for every (reasonably smooth) symmetric cylinder's profile and have finite energy, but are singular at two points on the strip boundary, representing the points where the free surface meets the cylinder's hull. The relevance of these results for the solvability of the nonlinear problem is discussed.

Article information

Adv. Differential Equations Volume 7, Number 4 (2002), 385-418.

First available in Project Euclid: 27 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 76B20: Ship waves
Secondary: 35J20: Variational methods for second-order elliptic equations 35J25: Boundary value problems for second-order elliptic equations 35Q35: PDEs in connection with fluid mechanics 76B03: Existence, uniqueness, and regularity theory [See also 35Q35] 76M30: Variational methods


Pierotti, Dario. The subcritical motion of a surface-piercing cylinder: existence and regularity of waveless solutions of the linearized problem. Adv. Differential Equations 7 (2002), no. 4, 385--418.

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